Properties and types
– The pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix.
– A double helix consists of two helices with the same axis, differing by a translation along the axis.
– A circular helix has constant band curvature and constant torsion.
– A conic helix, also known as a conic spiral, is a spiral on a conic surface.
– A curve is called a general helix or cylindrical helix if its tangent makes a constant angle with a fixed line in space.
Handedness
– Helices can be either right-handed or left-handed.
– Right-handed helix moves away from the observer in a clockwise screwing motion.
– Left-handed helix moves towards the observer in a clockwise screwing motion.
– Handedness is a property of the helix, not of the perspective.
– Right-handed and left-handed helices cannot be transformed into each other without using a mirror.
Mathematical description
– A helix is a curve in 3-dimensional space.
– The parametrization in Cartesian coordinates defines a helix.
– The simplest equations for a helix are cos(t), sin(t), and t.
– The helix traces a right-handed helix of pitch 2 and radius 1 about the z-axis.
– In cylindrical coordinates, the same helix is parametrized by r(t) = 1, θ(t) = t, and h(t) = t.
Arc length, curvature, and torsion
– A circular helix of radius a and slope has an arc length of T√(a^2 + b^2).
– The curvature of a circular helix is |a| / (a^2 + b^2).
– The torsion of a circular helix is b / (a^2 + b^2).
– A helix has constant non-zero curvature and torsion.
– The ratio of curvature to torsion is b / (a^2 + b^2).
Examples
– The nucleic acid double helix is an example of a double helix in molecular biology.
– The Corkscrew roller coaster at Cedar Point amusement park is an example of a conic helix.
– Some curves found in nature consist of multiple helices of different handedness joined together.
– Most hardware screw threads are right-handed helices.
– Pitch space in music is often modeled with helices or double helices.Sources: https://en.wikipedia.org/wiki/Helix
